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INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A29

ON P,Q-BINOMIAL COEFFICIENTS

Roberto B. CorcinoDepartment of Mathematics, Mindanao State University, Marawi City, Philippines 9700

[email protected]

Received: 10/8/07, Revised: 3/29/08, Accepted: 6/25/08, Published: 7/18/08

Abstract

In this paper, we develop the theory of a p, q-analogue of the binomial coe!cients. Someproperties and identities parallel to those of the usual and q-binomial coe!cients will beestablished including the triangular, vertical, and the horizontal recurrence relations, hor-izontal generating function, and the orthogonality and inverse relations. The constructionand derivation of these results give us an idea of how to handle complex computations in-volving the parameters p and q. This may be a good start in developing the theory ofp, q-analogues of some special numbers in combinatorics. Furthermore, several interestingspecial cases will be disclosed which are analogous to some established identities of the usualbinomial coe!cients.

Introduction

The binomial coe!cients, denoted by!

nk

", play an important role in enumerative combina-

torics. These numbers appear as the coe!cients in the expansion of the binomial expression(x + y)n. That is,

(x + y)n =n#

k=0

$n

k

%xkyn!k.

This identity is known as the Binomial Theorem [5][6][16][19]. If y = 1, this identity becomes

(x + 1)n =n#

k=0

$n

k

%xk (1)

which is the horizontal generating function for the binomial coe!cients. The explicit valueof the binomial coe!cients is given by

$n

k

%=

n!

k!(n! k)!, if 0 " k " n, and

$n

k

%= 0, otherwise.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A29 2

A more general definition of the binomial coe!cients is given in [11] where n can be a complexnumber; in particular the binomial theorem (1) holds in this more general setting if r # 0is an integer or |x/y| < 1. This can be interpreted as the number of possible k-subsets outof a set of n distinct elements or the number of ways to choose k elements from the set of ndistinct elements. The binomial coe!cients are also known as combinations or combinatorialnumbers.

Almost all books in combinatorics devote one chapter for the discussion of binomialcoe!cients ( see, e.g., [5] [6][16][19] ). This shows that the binomial coe!cients are alreadywell-studied and well-discussed in the past resulting to possession of numerous propertiesand identities. To mention a few, we have the triangular recurrence relation

$n + 1

k

%=

$n

k

%+

$n

k ! 1

%, (2)

the identity#

k even

$n

k

%=

#

k odd

$n

k

%, (3)

the inverse relation

fn =n#

k=0

$n

k

%gk $% gn =

n#

k=0

(!1)n!k

$n

k

%fk, (4)

Chu Shih-Chieh’s identity$

n + 1

k + 1

%=

$k

k

%+

$k + 1

k

%+ . . . +

$n

k

%,

and the orthogonality relation

n#

j=i

(!1)n!j

$n

j

%$j

i

%=

n#

j=i

(!1)j!i

$n

j

%$j

i

%= !ni =

&0 n &= i1 n = i

(5)

where !ni is called the Kronecker delta.

In constructing the properties and identities of some special numbers, binomial coe!-cients are frequently involved. That is why the q-analogue of the binomial coe!cients playsan important role in developing the theory of the q-analogue of these special numbers. From[3][6], the q-analogue of the binomial coe!cients

'nk

(q

is defined by

)n

k

*

q=

k+

i=1

qn!i+1 ! 1

qi ! 1,

)n

0

*

q= 1, q &= 1.

This q-analogue is also called the q-binomial coe!cient, a Gaussian coe!cient, or a Gaussianpolynomial. Using the notation

[k]q =qk ! 1

q ! 1, q &= 1,


we can rewrite the q-binomial coe!cient as)n

k

*

q=

[n]q!

[k]q![n! k]q!=

[n]q[k]q

,n! 1

k ! 1

-

q

where [k]q!

denotes the q-factorial which is given by [k]q! = [k]q[k!1]q . . . [2]q[1]q. It can easily be verifiedthat, when q ' 1, the q-binomial coe!cients turn into the usual binomial coe!cients.

Unlike the usual binomial coe!cients, the q-binomial coe!cients'

nk

(qhave only a limited

number of properties and identities. The following are some of the properties and identitiesfor

'nk

(q

which are taken from [6]. We have the triangular recurrence relation

)n

k

*

q=

,n! 1

k ! 1

-

q

+ qk

,n! 1

k

-

q

, (6)

the identity #

k even

)n

k

*

q=

#

k odd

)n

k

*

q, (7)

the inverse relation

fn =n#

k=0

)n

k

*

qgk $% gn =

n#

k=0

(!1)n!kq(n!k

2 ))n

k

*

qfk, (8)

and the orthogonality relation

n#

j=i

(!1)n!jq(n!j

2 ),n

j

-

q

,j

i

-

q

=n#

j=i

(!1)j!iq(j!i2 )

,n

j

-

q

,j

i

-

q

= !ni. (9)

Analogous to the expansion in (1), we have

n!1+

r=0

(1 + xqr) =n#

k=0

q(k2)

)n

k

*

qxk. (10)

Moreover, the q-binomial coe!cient'

m+nm

(q

may be interpreted as a polynomial in q whose

coe!cient in qk counts the number of distinct partitions of k elements which fit inside anm( n rectangle [1].

The p, q-analogue of the usual binomial coe!cient which is also called the p, q-binomialcoe!cient is defined by

)n

k

*

pq=

k+

i=1

pn!i+1 ! qn!i+1

pi ! qi, p &= q. (11)

Using the notation [k]pq = pk!qk

p!q we can rewrite (11) as follows

)n

k

*

pq=

[n]pq!

[k]pq![n! k]pq!=

[n]pq

[k]pq

,n! 1

k ! 1

-

pq


where [k]pq! = [k]pq[k ! 1]pq . . . [2]pq[1]pq. It may be worth noticing that

)n

k

*

pq=

k+

i=1

qn!i+1 ! pn!i+1

qi ! piand

)n

k

*

pq=

[n]pq!

[n! k]pq![n! (n! k)]pq!.

Hence, )n

k

*

pq=

)n

k

*

qpand

)n

k

*

pq=

,n

n! k

-

pq

. (12)

Clearly, when p = 1, the p, q-binomial coe!cient'

nk

(pq

reduces to the q-binomial coe!cient'nk

(q.

In this paper, we develop the theory of p, q-analogue of the binomial coe!cients byestablishing some properties and identities parallel to those in (6) - (10). Some special caseswill be disclosed which somehow give a clearer description of the p, q-binomial coe!cients.

Recurrence Relations and Generating Function

For a quick computation of the first values of p, q-binomial coe!cients, we have the followingtriangular recurrence relations which are of similar form with those in (2) and (6).

Theorem 1. The p, q-binomial coe!cients satisfy the following triangular recurrence rela-tions ,

n + 1

k

-

pq

= pk)n

k

*

pq+ qn!k+1

,n

k ! 1

-

pq

(13)

,n + 1

k

-

pq

= qk)n

k

*

pq+ pn!k+1

,n

k ! 1

-

pq

(14)

with initial conditions'

00

(pq

= 1,'

nn

(pq

= 1, and'

nk

(pq

= 0 for k > n.

Proof. Evaluating the right-hand side of (13), we get

pk)n

k

*

pq+ qn!k+1

,n

k ! 1

-

pq

= pkk+

i=1

pn!i+1 ! qn!i+1

pi ! qi+ qn!i+1

k!1+

i=1

pn!i+1 ! qn!i+1

pi ! qi

=

&.k!1i=1 (pn!i+1 ! qn!i+1

/0pk(pn!k+1 ! qn!k+1 + qn!k+1(pk ! qk)

1

.ki=1(p

i ! qi)

=

.ki=1(p

(n+1)!i+1 ! q(n+1)!i+1).k

i=1(pi ! qi)

=

,n + 1

k

-

pq

.

Note that we can rewrite (13) as'

n+1k

(qp

= qk'

nk

(qp

+ pn!k+1'

nk!1

(qp

. Using (12), we obtain

(14). !


Taking p = 1, (14) yields'

n+1k

(q

= qk'

nk

(q+

'n

k!1

(q

which is precisely equivalent to (6).

Similarly, when p = 1, (13) gives'

n+1k

(q

='

nk

(q+ qn!k+1

'n

k!1

(q

which is another form of the

triangular recurrence relation for q-binomial coe!cients.

Using Theorem 1, we can quickly generate the first values of'

nk

(pq

as shown in thefollowing table.

n/k 1 2 3 4 51 12 p + q 13 p2 + pq + q2 p2 + pq + q2 14 p3 + p2q + pq2 (p2 + pq + q2)(p2 + q2) p3 + p2q + pq2 + q3 1

+q3

5 p4 + p3q + p2q2 p2(p2 + pq + q2)(p2 + q2) p3(p3 + p2q + pq2 + q3) p4 + p3q + p2q2 1+pq3 + q4 +q3(p3 + p2q + pq2 + q3) +q2(p2 + pq + q2)(p2 + q2) +pq3 + q4

Table 1. Table of Values for'

nk

(pq

One may try to verify (12) for n = 1, 2, 3, 4, 5 using Table 1. In fact, we can easily see it ifwe write the entries of Table 1 in the following way:

1

1 1

1 p + q 1

1 p2 + pq + q2 p2 + pq + q2 1

1 p3 + p2q + pq2 + q3 (p2 + pq + q2)(p2 + q2) p3 + p2q + pq2 + q3 1

1 p4 + p3q + p2q2 p2(p2 + pq + q2)(p2 + q2) p3(p3 + p2q + pq2 + q3) p4 + p3q + p2q2 1

+pq3 + q4 + q3(p3 + p2q + pq2 + q3) + q2(p2 + pq + q2)(p2 + q2) + pq3 + q4

This figure is analogous to Pascal’s Triangle of the usual binomial coe!cients. As we cansee, it possesses the same symmetry as Pascal’s Triangle. However, the identities that wehave for the p, q-binomial coe!cients may not exactly be of the same structure as those inthe usual binomial coe!cients.

Note that if we apply (13) thrice to'

n+1k+1

(pq

, we obtain


,n + 1

k + 1

-

pq

= qn!(k+1)+1)n

k

*

pq+ pk+1

,n

k + 1

-

pq

= qn!k)n

k

*

pq+ pk+1

2qn!k!1

,n! 1

k

-

pq

+ pk+1

,n! 1

k + 1

-

pq

3

= qn!k)n

k

*

pq+ pk+1qn!k!1

,n! 1

k

-

pq

+ p2(k+1)

2qn!k!2

,n! 2

k

-

pq

+ pk+1

,n! 2

k + 1

-

pq

3

= qn!k)n

k

*

pq+ pk+1qn!k!1

,n! 1

k

-

pq

+ p2(k+1)qn!k!2

,n! 2

k

-

pq

+ p3(k+1)

,n! 2

k + 1

-

pq

.

Continuing this process until the (n! k)th application of (13), we get,n + 1

k + 1

-

pq

= qn!k)n

k

*

pq+ pk+1qn!k!1

,n! 1

k

-

pq

+ p2(k+1)qn!k!2

,n! 2

k

-

pq

+ p3(k+1)qn!k!3

,n! 3

k

-

pq

+ · · · + p(n!k)(k+1)

,k

k

-

pq

.

This is a vertical recurrence relation for the p, q-binomial coe!cients. On the other hand, ifwe rewrite (14) as

)n

k

*

pq= p!(n!k)

,n + 1

k + 1

-

pq

! p!(n!k)qk+1

,n

k + 1

-

pq

and apply this recurrence relation to itself (n! k) times, we obtain a horizontal recurrencerelation for the p, q-binomial coe!cients. Let us state these results in the following theorem.

Theorem 2. The p, q-binomial coe!cients satisfy the following vertical recurrence relation

,n + 1

k + 1

-

pq

=n#

j=k

p(n!j)(k+1)qj!k

,j

k

-

pq

and horizontal recurrence relation

)n

k

*

pq=

n!k#

j=0

(!1)jp!(j+1)(n!k)+(j+12 )qjk+(j+1

2 ),

n + 1

k + j + 1

-

pq

.


For example, using the vertical recurrence relation in Theorem 2, we can compute'

43

(pq

as follows:,4

3

-

pq

= p(3!2)(2+1)q2!2

,2

2

-

pq

+ p(3!3)(2+1)q3!2

,3

2

-

pq

= p3 + q(p2 + pq + q2) = p3 + p2q + pq2 + q3.

This is exactly the value of'

43

(pq

that appears in Table 1. One may also try to compute'43

(pq

using the horizontal recurrence relation.

The vertical and horizontal recurrence relations in Theorem 2 may be considered p, q-analogues of the Hockey Stick identities, which are also known as Chu Shih-Chieh’s identities.More precisely, if we let p = 1, Theorem 2 will give the following vertical and horizontalrecurrence relations for the q-binomial coe!cients, respectively,

,n + 1

k + 1

-

q

=n#

j=k

qj!k

,j

k

-

q

and)n

k

*

q=

n!k#

j=0

(!1)jqjk+(j+12 )

,n + 1

k + j + 1

-

q

.

Furthermore, when q ' 1, the former will reduce to Chu Shih-Chieh’s identity as given inthe introduction and the latter will reduce to

$n

k

%=

$n + 1

k + 1

%!

$n + 1

k + 2

%+ . . . + (!1)n!k

$n + 1

n + 1

%

which is another recurrence relation for the usual binomial coe!cients.

The vertical and horizontal recurrence relations in Theorem 2 can easily be rememberedwith the help of the pattern as shown in the table below.

n / k . . . k k + 1 k + 2 . . . n + 1

... ) )) )

k)

kk

*

pq

) )) )

k + 1)

k+1k

*

pq

) )... )

... )) )

n' n

k

(pq

* +* +* +* ,!,!,!,! ,!,!,!,! ,! ,!,!,!,!

n + 1 *)

n+1k+1

*

pq

)n+1k+2

*

pq. . .

)n+1n+1

*

pq

,!,!,!,! ,!,!,!,! ,! ,!,!,!,!


The entries involved in the table to compute the value of'

n+1k+1

(pq

and'

nk

(pq

using the vertical

and horizontal recurrence relations in Theorem 2 clearly form a hockey stick as illustratedby the arrows in the table.

We observe that

1 + x =

,1

0

-

pq

+

,1

1

-

pq

x

(1 + x)(p + qx) = p + (p + q)x + qx2

= p

,2

0

-

pq

+

,2

1

-

pq

x + q

,2

2

-

pq

x2

and

(1 + x)(p + qx)(p2 + q2x) = p3 + p(p2 + pq + q2)x + q(p2 + pq + q2)x2 + q3x3

= p3

,3

0

-

pq

+ p

,3

1

-

pq

x + q

,3

2

-

pq

x2 + q3

,3

3

-

pq

x3.

This observation gives us an idea that we can also construct a horizontal generating functionfor the p, q-binomial coe!cients which is analogous to those in (1) and (6). The next theoremcontains the horizontal generating function mentioned above.

Theorem 3. The horizontal generating function for the p, q-binomial coe!cients is givenby

n!1+

r=0

(pr + xqr) =n#

k=0

p(n!k

2 )q(k2)

)n

k

*

pqxk.

Proof. We prove this by induction on n. For n = 1, 2, 3, the generating function is alreadyverified above. Suppose it is true for some n # 1, that is,

n!1+

r=0

(pr + xqr) =n#

k=0

p(n!k

2 )q(k2)

)n

k

*

pqxk.

Then, using the inductive hypothesis, we have

n+

r=0

(pr + xqr) = (pn + xqn)n#

k=0

p(n!k

2 )q(k2)

)n

k

*

pqxk

=n#

k=0

p(n!k

2 )+nq(k2)

)n

k

*

pqxk +

n#

k=0

p(n!k

2 )q(k2)+n

)n

k

*

pqxk+1

=n+1#

k=0

4p(

n!k2 )+nq(

k2)

)n

k

*

pq+ p(

n!k+12 )q(

k!12 )+n

,n

k ! 1

-

pq

5xk

=n+1#

k=0

p(n!k+1

2 )q(k2)

4pk

)n

k

*

pq+ qn!k+1

,n

k ! 1

-

pq

5xk.


Applying Theorem 1, we getn+

r=0

(pr + xqr) =n+1#

k=0

p(n!k+1

2 )q(k2)

,n + 1

k

-

pq

xk. !

As a direct consequence of Theorem 3, we have the following corollary which contains anidentity analogous to those in (3) and (7).

Corollary 1. For n # 1, we have#

k even

p(n!k

2 )q(k2)

)n

k

*

pq=

#

k odd

p(n!k

2 )q(k2)

)n

k

*

pq. (15)

Proof. With x = !1, Theorem 3 givesn!1+

r=0

(pr ! qr) =n#

k=0

(!1)kp(n!k

2 )q(k2)

)n

k

*

pq.

Since pr ! qr = 0 when r = 0, we haven#

k=0

(!1)kp(n!k

2 )q(k2)

)n

k

*

pq= 0.

This is precisely equivalent to (15). !

The next corollary can easily be deduced from Theorem 3 by taking x = 1.

Corollary 2. For n # 1, we have

n#

k=0

67

8p(

n!k2 )q(

k2)

'nk

(pq.n!1

r=0 (pr + qr)

9:

; = 1. (16)

Corollary 2 may be interpreted as follows. Consider an experiment with " as the samplespace such that

|"| =n!1+

r=0

(pr + qr),

and a random variable X that takes the value from 0 to n which determines the outcomeof the experiment. Suppose that a given event Ek - " is associated with the value X = ksuch that

|Ek| = p(n!k

2 )q(k2)

)n

k

*

pq.

Then (16) shows that

Pr(X = k) =p(

n!k2 )q(

k2)

'nk

(pq.n!1

r=0 (pr + qr)

is a probability distribution of X.

The horizontal generating function in Theorem 3 may be a good tool in giving a combi-natorial interpretation of the p, q-binomial coe!cients. One may try to use the method ofgenerating functions as discussed in [5],[16], and [19].


Orthogonality and Inverse Relations

Inverse relations are useful tools in deriving necessary formulas of some special numbers incombinatorics. The inverse relations in (4) and (8) were used many times in transformingsome identities into a completely di#erent and interesting form. For instance, the second formof the horizontal recurrence relation for the generalized Stirling numbers [7] was obtainedby transforming the explicit formula of the generalized Stirling numbers using the inverserelation in (4), and the explicit formula of the q-analogue for the generalized Stirling numbers[8] was transformed into a horizontal recurrence relation using (8). Thus, for possible use indeveloping the theory of p, q-analogue of some special numbers, an inverse relation for thep, q-binomial coe!cients must also be established.

The following theorem contains the orthogonality relation for the p, q-binomial coe!cientwhich is necessary in establishing the inverse relation.

Theorem 4. The orthogonality relation for the p, q-binomial coe!cients is given by

n#

j=i

(!1)n!jp(n!j

2 )q(j!i2 )

,n

j

-

pq

,j

i

-

pq

=n#

j=i

(!1)j!ip(j!i2 )q(

n!j2 )

,n

j

-

pq

,j

i

-

pq

= !ni.

Proof. Clearly, it is true for n = 0, 1. Suppose that for some n # 0

Hn =n#

j=i

(!1)n!jp(n!j

2 )q(j!i2 )

,n

j

-

pq

,j

i

-

pq

= !ni.

Then, by Theorem 1, we have

Hn+1 =n+1#

j=i

(!1)n+1!jp(n+1!j

2 )q(j!i2 )pj

,n

j

-

pq

,j

i

-

pq

+n+1#

j=i

(!1)n+1!jp(n+1!j

2 )q(j!i2 )qn!j+1

,n

j ! 1

-

pq

,j

i

-

pq

.

Using the facts that,

n

n + 1

-

pq

= 0, p(n+1!j

2 )+j = p(n!j

2 )pn, and q(j!i2 )+(n!j+1) = q(

j!1!i2 )qn!i,

we obtain

Hn+1 = (!1)pn!ni + qn!in#

j=i!1

p(n!j

2 )q(j!i2 )

,n

j

-

pq

,j + 1

i

-

pq

.

Again, by applying Theorem 1, we get

Hn+1 = !pn!ni + qn!ipi!ni + qn!in#

j=i!1

p(n!j

2 )q(j!i2 )

,n

j

-

pq

qj!i+1

,j

i! 1

-

pq


= (!pn + qn!ipi)!ni + qn!iqn#

j=i!1

p(n!j

2 )q(j!(i!1)

2 ),n

j

-

pq

,j

i! 1

-

pq

= 0 + qn!i+1!n,i!1 = !n+1,i.

This proves the first form of the orthogonality relation.

To prove the other form, we first interchange p and q to obtain

n#

j=i

(!1)j!ip(j!i2 )q(

n!j2 )

,n

j

-

pq

,j

i

-

pq

= (!1)n!in#

j=i

(!1)n!jp(n!j

2 )q(j!i2 )

,n

j

-

qp

,j

i

-

qp

.

Using (12) and the first form of the orthogonality relation, we have

n#

j=i

(!1)j!ip(j!i2 )q(

n!j2 )

,n

j

-

pq

,j

i

-

pq

= (!1)n!in#

j=i

(!1)n!jp(n!j

2 )q(j!i2 )

,n

j

-

pq

,j

i

-

pq

= (!1)n!i!ni = !ni. !

The first form of the orthogonality relation in Theorem 4 may be written as<n

j=i anjbji =!ni where

anj = (!1)n!jp(n!j

2 ),n

j

-

pq

and bji = q(j!i2 )

,j

i

-

pq

.

If we let M and =M be two matrices such that M = [aij]0"i,j"m and =M = [bij]0"i,j"m, then

M =M = [!ij]0"i,j"m = Im. That is, =M = M!1 and M = =M!1.

The next theorem is a kind of transformation from the identity in Theorem 3 that inter-changes the role of xk and

.n!1r=0 (qr + xpr) in the formula.

Theorem 5. The p, q-binomial coe!cients satisfy the following relation

p(n2)xn =

n#

j=0

(!1)n!jp(n!j

2 ),n

j

-

pq

j!1+

r=0

(qr + xpr).

Proof. Clearly, p(n2)xn =

n#

i=0

!nip(i2)xi. Then, using Theorem 4, we have

p(n2)xn =

n#

i=0

4n#

j=i

(!1)n!jp(n!j

2 )q(j!i2 )

,n

j

-

pq

,j

i

-

pq

5p(

i2)xi

=n#

j=0

(!1)n!jp(n!j

2 ),n

j

-

pq

4j#

i=0

q(j!i2 )p(

i2)

,j

i

-

pq

xi

5.

Applying Theorem 3 completes the proof of the theorem. !


We notice that when x = 0, Theorem 5 givesn#

j=0

(!1)n!jp(n!j

2 )q(j2)

,n

j

-

pq

= 0, which is

precisely equivalent to Corollary 1. On the other hand, when x = 1, Theorem 5 yieldsn#

j=0

(!1)n!jp(n!j

2 )!(n2)

,n

j

-

pq

j!1+

r=0

(qr + pr) = 1

which consequently givesn#

j=1

(!1)n!jp(n!j

2 )!(n2)

,n

j

-

pq

j!1+

r=0

(qr + pr) = 0 if n is even.

That means

#

j even(#=0)

p(n!j

2 )!(n2)

,n

j

-

pq

j!1+

r=0

(qr + pr) =#

j odd

p(n!j

2 )!(n2)

,n

j

-

pq

j!1+

r=0

(qr + pr)

if n is even.

With the aid of Theorem 4, we can easily prove the inverse relation for the p, q-binomialcoe!cients which is given in the following theorem.

Theorem 6. The inverse relation for the p, q-binomial coe!cients is given by

fn =n#

j=0

(!1)n!jp(n!j

2 ),n

j

-

pq

gj $% gn =n#

j=0

q(n!j

2 ),n

j

-

pq

fj.

Proof. (%) Using the given hypothesis, we have

n#

j=0

q(n!j

2 ),n

j

-

pq

fj =n#

j=0

j#

i=0

(!1)j!ip(j!i2 )q(

n!j2 )

,n

j

-

pq

,j

i

-

pq

gi

=n#

i=0

4n#

j=i

(!1)j!ip(j!i2 )q(

n!j2 )

,n

j

-

pq

,j

i

-

pq

5gi.

Applying the second form of the orthogonality relation in Theorem 4, we obtainn#

j=0

q(n!j

2 ),n

j

-

pq

fj =n#

j=0

!nigi = gn.

($) This can easily be done by following the same argument as above and applying the firstform of the orthogonality relation. !

If we set fn = p(n2)xn and gj =

.j!1r=0(q

r + xpr), then, using Theorem 6, the identity inTheorem 5 becomes

n!1+

r=0

(qr + xpr) =n#

j=0

q(n!j

2 )p(j2)

,n

j

-

pq

xj

which gives the expansion in Theorem 3 by interchanging p and q.


Some Remarks and Related Literature

There are still many properties and identities that we need to establish for the p, q-binomialcoe!cients which are parallel to those in the usual binomial coe!cients. For instance,Vandermonde’s identity, which is given by

$m + n

k

%=

k#

r=0

$m

r

%$n

k ! r

%, (17)

can be obtained using the Binomial Theorem by multiplying the expansions (x + 1)m and(x+1)n to get the coe!cients of xk which turns out to be the right-hand side of (17), whichis then compared to the coe!cients of xk in the expansion of (x + 1)m+n.

The corresponding q-analogue of Vandermonde’s identity is also known as the q-Vander-monde identity [10], which is given by

,m + n

k

-

q

=k#

r=0

qr(m!k+r))m

r

*

q

,n

k ! r

-

q

,

and can be derived by paralleling what has been done to obtain (17), but this time usingthe identity in q-binomial theory

(A + B)m(A + B)n = (A + B)m+n

with operators A and B that q-commute (i.e., BA = qAB). With these facts, one may tryto construct a p, q-analogue of Vandermonde’s identity by defining some operators analogousto those operators that q-commute.

The q-binomial coe!cients were included in the classical families of numbers that ariseas a particular case of U -Stirling numbers by Medicis and Leroux [9]. More precisely, theq-binomial coe!cients can be expressed as

)n

k

*

q=

#

!$TA(k,n!k)

+

c$!

q|c| (18)

where TA(k, n ! k) denotes the set of A-tableaux ( defined in [9] ) with exactly n ! kcolumns whose lengths are in the set {0, 1, 2, . . . , k}. This formula may give a combinatorialinterpretation of the q-binomial coe!cients in the context of 0-1 tableau as defined in [9].Furthermore, identity (18) may be written as

)n

k

*

q=

#

0"j1"j2"..."jn!k"k

n!k+

i=1

qji =#

0"j1"j2"..."jn!k"k

qj1+j2+...+jn!k .

If we can express the p, q-binomial coe!cients parallel to this expression, then the p, q-binomial coe!cients may also be given a combinatorial interpretation in the context of 0-1tableau.


In the study of (p, q)-hypergeometric series r$s [10][13] which is defined by

r$s((a1p, a1q), . . . , (arp, arq); (b1p, b1q), . . . , (bsp, bsq); (p, q), z)

=%#

n=0

((a1p, a1q), . . . , (arp, arq); (p, q))n

((b1p, b1q), . . . , (bsp, bsq); (p, q))n((p, q); (p, q))n

!(!1)n(q/p)n(n!1)/2

"1+s!rzn

where |q/p| < 1 and

((x1p, x1q), . . . , (xrp, xrq); (p, q))n =r+

i=1

((xip, xiq); (p, q))n

with ((xip, xiq); (p, q))n =.n!1

m=0(xippm ! xiqqm), the p, q-binomial coe!cients appeared insome of its identities. As mentioned in [13], when r = 1 and s = 0, the above (p, q)-hypergeometric series gives

1$0((pn, qn);!; (p, q), z) =

%#

k=0

,n! 1 + k

k

-

pq

=

4n#

k=0

)n

k

*

pq(pq)k(k!1)/2(!z)k

5!1

(18)

and this can further be written as

1$0((pn, qn);!; (p, q), z) =

4n#

k=0

)n

k

*

q!1q(!z)k

5!1

with )n

k

*

q!1q=

((q!1, q); (q!1, q))n

((q!1, q); (q!1, q))k((q!1, q); (q!1, q))n!k

which should be relevant in the context of quantum groups. Now, taking a = p"p andb = pn"q with z = "q/"p, we can rewrite (18) as

((a, b); (p, q))n =n#

k=0

)n

k

*

pq(!1)kp(

n!k2 )q(

k2)an!kbk. (19)

When a = 1, (19) gives

n!1+

i=0

(pi ! bqi) =n#

k=0

p(n!k

2 )q(k2)

)n

k

*

pq(!b)k,

which can also be obtained from Theorem 3 by taking x = !b.

The (p, q)-hypergeometric series is a generalization of the basic hypergeometric series orthe q-hypergeometric series which is given by

r#s(a1, a2, . . . , ar; b1, b2, . . . , bs; q, z) =%#

n=0

(a1, a2, . . . , ar; q)n

(b1, b2, . . . , bs; q)n(q; q)n

!(!1)nqn(n!1)/2

"1+s!rzn,


such that when a1p = a2p = . . . = arp = b1p = b2p = . . . = bsp = 1, a1q = a1, a2q =a2, . . . , arq = ar, b1q = b1, b2q = b2, . . . , bsq = bs, and p = 1, r$s ' r#s. Another form of ageneralization of q-hypergeometric series is the bibasic hypergeometric series [10] with twobases q and q1, which is given by

F (a, c; b, d; q, q1, z) =%#

n=0

(a; q)n(c; q1)n

(a; q)n(c; q1)n(q; q)n

!(!1)nqn(n!1)/2

"1+s!r>(!1)nqn(n!1)/2

1

?s1!r1

zn,

where

(a; q)n =r+

i=1

(ai; q)n, (ai; q)n =(ai; q)%

(aiqn; q)%, (ai; q)% =

%+

k=0

(1! aqk).

In this series, the two unconnected bases q and q1 are regarded and assigned partially todi#erent numerator and denominator parameters whereas in the (p, q)-hypergeometric seriesthe parameters p and q are inseparable and assigned to all the numerator and denominatorparameter doublets. As we can see in [13], it is very interesting and profitable to study the(p, q)-hypergeometric series as well as the p, q-binomial coe!cients. It is worth mentioningthat there is research related to physics which considers identities of p, q-binomial coe!cients[12]. For instance, Katriel and Kibler [14] defined the p, q-binomial coe!cients and derived ap, q-binomial theorem while discussing normal ordering for deformed boson operators obeying(p, q)-oscillator algebra introduced by R. Chakrabarti and R. Jagannathan in [4]. Smirnovand Wehrhahn [18] expressed the expansion of

!qJ0(1)J±(2) + J±(1)p!J0(2)

"l

in terms of the p, q-binomial coe!cients, where {J0(1), J±(1)} and {J0(2), J±(2)} are thegenerators of two commuting (p, q)-angular momentum algebras.

There are algorithmic approaches which can prove multi-basic hypergeometric seriesidentities. Zeilberger’s creative telescoping paradigm in the algebraic setting of di#er-ence fields [17] is one of these algorithms which computes a recurrence for the given sumS(n) =

<mk=0 f(n, k) where m might depend linearly on n, by finding c0(n), . . . , c!(n) and

g(n, k) such that g(n, k+1)!g(n, k) = c0(n)f(n, k)+. . .+c!(n)f(n+!, k). Another is Gosper’ssummation algorithm which finds a hypergeometric closed form of an indefinite sum of hyper-geometric terms, if such a closed form exists. This algorithm was extended by Bauer in [2] tothe case when the terms are simultaneously hypergeometric and multibasic hypergeometric.Riese[15] also presented an algebraically motivated generalization of Gosper’s algorithm toindefinite bibasic hypergeometric summation. Exploiting these algorithms might contributeto find and prove additional identities of the p, q-binomial coe!cients.

Acknowledgement. The author is grateful for the corrections and suggestions of theanonymous referee.


References

[1] G. E. Andrews, The Theory of Partitions, Chapt. 3, Cambridge University Press, 1998.

[2] A. Bauer and M. Petkovsek, Multibasic and Mixed Hypergeometric Gosper-Type Algorithm, J. SymbolicComput., 28(1999): 711-736.

[3] L. Carlitz, q-Bernoulli Numbers and Polynomials, Duke Math. J., 15(1948): 987-1000.

[4] R. Chakrabarti and R. Jagannathan, A (p, q)-Oscillator Realization of Two-parameter Quantum Alge-bras, J. Phys. A: Math. Gen. 24(1991) L711.

[5] C.C. Chen and K.M. Kho, Principles and Techniques in Combinatorics, World Scientific Publishing Co.Pte. Ltd., 1992.

[6] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, The Netherlands, 1974.

[7] R.B. Corcino, Some Theorems on Generalized Stirling Numbers, Ars Combinatoria, 60(2001): 273-286.

[8] R.B. Corcino, L.C. Hsu, and E.L. Tan, A q-Analogue of Generalized Stirling Numbers, The FibonacciQuarterly, 44(2006): 154-165.

[9] A. De Medicis and P. Leroux, Generalized Stirling Numbers, Convolution Formulas and p, q-Analogues,Can. J. Math, 47(1995): 474-499.

[10] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Math. Appl. 96 (SecondEdition), Cambridge Univ. Press, Cambridge, 2004.

[11] R. L. Graham, D.E. Knuth, and O. Patashnik, Concrete Mathematics:A Foundation for ComputerScience (Second Edition), Addison-Wesley, Reading, Massachusetts, 1989.

[12] R. Jagannathan, (P,Q)-Special Functions, Special Functions and Di!erential Equations, Proceedingsof a Workshop held at The Institute of Mathematical Sciences, Madras, India, January 13-24, 1997.

[13] R. Jagannathan and K. Srinivasa Rao, Two-Parameter Quantum Algebras, Twin-Basic Numbers, andAssociated Generalized Hypergeometric Series, To appear in the Proceedings of the International Con-ference on Number Theory and Mathematical Physics December 2005.

[14] J. Katriel and M. Kibler, Normal Ordering for Deformed Boson Operators and Operator-Valued De-formed Stirling Numbers, J. Phys. A: Math. Gen. 25(1992) 2683.

[15] A. Riese, A Generalization of Gosper’s Algorithm to Bibasic Hypergeometric Summation, Electron. J.Combin., 3(1996): 1-16 #R19.

[16] J. Riordan, An Introduction to Combinatorial Analysis, John Wiley and Son, Inc., 1958.

[17] C. Schneider, Symbolic Summation Assists Combinatorics, Sem. Lothar. Combin., 56(2007): 1-36.

[18] Y. Smirnov and R. Wehrhahn, The Clebsch-Gordan Coe!cients for the Two-Parameter QuantumAlgebra SUp,q(2) in the Lowdin-Shapiro Approach, J. Phys. A: Math. Gen. 25(1992) 5563.

[19] R. P. Stanley, Enumerative Combinatorics, Wadsworth and Brooks/Cole, Monterey, 1986.

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